So, having enchanting would increase my exp on making potions???
yes... you can enchant ring, amulet, falmer helmet, circlet, and gloves with fortify alchemy and equip them all at the same time. At 100 enchanting, this would boost the spell potency 125%, giving you 125% more exp gain and 125% more potion value per potion created. However at 100 enchanting alchemy is not 'really' needed except as a source of cash. Yes you can boost your stats a little more, but it's not huge by any means. Still it's easy and quick when done properly.
On another note, if you have enchanting high enough to do at least 22% reduction in spell cost, but preferably closer to 25%, you should enchant 4 items with conjuration reduction cost and try getting your conjuration up to 100, and then buying summon dremora lord spells (from the college of winterhold) . You will be very powerful and be able to go most places safely even on master. This will allow you to hit the area that has hundreds of those eggs, which is probably the fastest way combined with catching lunar moths which is also easy and fast, to level alchemy quickly. Conjuration is also just plain fun. (with dremoras, the other stuff is underpowered on master)

Dual casting soul trap on corpses level up the skill quick.